Sensitivity functions for single-input linear time-invariant systems

Abstract

Frequency domain techniques are utilized to establish general symmetry and simultaneity properties for sensitivity functions of the phase canonical form of a single-input, linear, time-invariant n th order controllable system. It is demonstrated that the computation of the l th sensitivity function, defined by the (l + 1) th tensor: β≜ ∂ lz i ∂α jl∂α ... jl-1∂α j 2 ∂α j 1 ∣ α=α 0 for i = 1,2,…,n, j k = 1,2,…,n, k = 1,2,…,l, can be performed iteratively and requires only one n th-order model in addition to the (l - 1) th sensititvity function. This Complete Simultaneity Property is further shown to apply to the generation of sensitivity functions of arbitrary n th-order, single-input systems. Implementation of this general result needs only (l + 1)n integrators. Hence, it provides a marked improvement over sensitivity point analysis which employs [ n + Σ l k=1 n k ] integrators for an n th order system represented in phase-canonical form. The application of the technique to general single-input, linear, time-invariant systems is indicated and the extension of the approach to multi-input, linear, time-invariant systems is mentioned. A scheme is provided which can be implemented on an analog or digital computer to study and evaluate the sensitivity of single-input, linear, time-invariant systems. Examples illustrate both the key ideas and application of the results.